FOURIER SERIES Philip Hall Jan 2011 Definition of a Fourier series A Fourier series may be defined as an expansion of a function in a series of sines and cosines such as (7.1) The coefficients are related to the periodic function f(x) by definite integrals: Eq.(7.11) and (7.12) to be mentioned later on. Henceforth we assume f satisfies the following (Dirichlet) conditions: (1) f(x) is a periodic function; (2) f(x) has only a finite number of finite discontinuities; (3) f(x) has only a finite number of extrem values, maxima and minima in the interval [0,2p]. Fourier series are named in honour of Joseph Fourier (1768-1830), who made important contributions to the study of trigonometric series, in connection with the solution of the heat equation

(7.3) Expressing cos nx and sin nx in exponential form, we may rewrite Eq.(7.1) as (7.4) in which (7.5) and (7.6)

Completeness and orthogonality We can show Fourier series satisfy certain completeness properties- see next year Now we need to find out how to determine the coefficients in the Fourier series expansion and find out in what circumstances the Fourier series and function coincide..

We can easily check the orthogonal relation for different values of the eigenvalue n by choosing the interval (7.7) (7.8) for all integer m and n. (7.9)

By use of these orthogonality, we are able to obtain the coefficients (7.9) Similarly (7.10) (7.11) (7.12) Substituting them into Eq.(7.1), we write

EXAMPLE: Sawtooth wave (7.13) This equation offers one approach to the development of the Fourier integral and Fourier transforms. EXAMPLE: Sawtooth wave Let us consider a sawtooth wave (7.14) For convenience, we shall shift our interval from to . In this interval we have simply f(x)=x. Using Eqs.(7.11) and (7.12), we have

So, the expansion of f(x) reads (7.15) . Figure 7.1 shows f(x) for the sum of 4, 6, and 10 terms of the series. Three features deserve comment. There is a steady increase in the accuracy of the representation as the number of terms included is increased. 2.All the curves pass through the midpoint at

Figure 7.1 Fourier representation of sawtooth wave

CONVERGENCE OF FOURIER SERIES W assume f satisfies the following (Dirichlet) conditions: (1) f(x) is a periodic function; (2) f(x) has only a finite number of finite discontinuities; (3) f(x) has only a finite number of extrem values, maxima and minima in the interval [0,2p]. We can then show that at a point where f(x) is continuous the Fourier series converges to f(x). However at a point of discontinuity the Fourier series converges to ½(sum of right and left hand limits of limits of f(x). Hence at any point x (7.18) where of course the right and left hand limits coincide where f is continuous.

7.2 ADVANTAGES, USES OF FOURIER SERIES Discontinuous Function One of the advantages of a Fourier representation over some other representation, such as a Taylor series, is that it may represent a discontinuous function. An example id the sawtooth wave in the preceding section. Other examples are considered in Section 7.3 and in the exercises. Periodic Functions Related to this advantage is the usefulness of a Fourier series representing a periodic functions . If f(x) has a period of , perhaps it is only natural that we expand it in , a series of functions with period , , This guarantees that if our periodic f(x) is represented over one interval or the representation holds for all finite x.

is an even function of x. Hence , At this point we may conveniently consider the properties of symmetry. Using the interval , is odd and is an even function of x. Hence , by Eqs. (7.11) and (7.12), if f(x) is odd, all if f(x) is even all . In other words, even, (7.21) odd. (7.21) Frequently these properties are helpful in expanding a given function. We have noted that the Fourier series periodic. This is important in considering whether Eq. (7.1) holds outside the initial interval. Suppose we are given only that (7.23) and are asked to represent f(x) by a series expansion. Let us take three of the infinite number of possible expansions.

1.If we assume a Taylor expansion, we have (7.24) a one-term series. This (one-term) series is defined for all finite x. 2.Using the Fourier cosine series (Eq. (7.21)) we predict that (7.25) 3.Finally, from the Fourier sine series (Eq. (7.22)), we have (7.26)

Figure 7.2 Comparison of Fourier cosine series, Fourier sine series and Taylor series.

These three possibilities, Taylor series, Fouries cosine series, and Fourier sine series, are each perfectly valid in the original interval . Outside, however, their behavior is strikingly different (compare Fig. 7.3). Which of the three, then, is correct? This question has no answer, unless we are given more information about f(x). It may be any of the three ot none of them. Our Fourier expansions are valid over the basic interval. Unless the function f(x) is known to be periodic with a period equal to our basic interval, or th of our basic interval, there is no assurance whatever that representation (Eq. (7.1)) will have any meaning outside the basic interval. It should be noted that the set of functions , , forms a complete orthogonal set over . Similarly, the set of functions , forms a complete orthogonal set over the same interval. Unless forced by boundary conditions or a symmetry restriction, the choice of which set to use is arbitrary.

FOURIER SERIES OVER ARBITRARY INTERVALS So far attention has been restricted to an interval of length of . This restriction may easily be relaxed. If f(x) is periodic with a period , we may write (7.27) with (7.28) (7.29)

replacing x in Eq. (7.1) with and t in Eq. (7.11) and (7.12) with (For convenience the interval in Eqs. (7.11) and (7.12) is shifted to . ) The choice of the symmetric interval (-L, L) is not essential. For f(x) periodic with a period of 2L, any interval will do. The choice is a matter of convenience or literally personal preference.

HALF RANGE FOURIER SERIES We have seen above that even and odd functions have Fourier series With either or zero. Now suppose we have a function defined on an interval [0,K], then we extend it as an even or odd function of period K so as to produce a Fourier cosine or sine series. But note the period is now 2K, it is always wise to sketch the extended function. Exercise: extend the functions given below as odd and even functions, sketch the extended functions and give the formulae for the Fourier coefficients: (7.4-5) Note that extending a function to produce either an odd or even function can lead to discontinuities.

7.3 APPLICATION OF FOURIER SERIES ) 7.3 APPLICATION OF FOURIER SERIES Example 7.3.1 Square Wave ——High Frequency One simple application of Fourier series, the analysis of a “square” wave (Fig. (7.5)) in terms of its Fourier components, may occur in electronic circuits designed to handle sharply rising pulses. Suppose that our wave is designed by

(7.30) From Eqs. (7.11) and (7.12) we find (7.31) (7.32) (7.33)

Example 7.3.2 Full Wave Rectifier The resulting series is (7.36) Except for the first term which represents an average of f(x) over the interval all the cosine terms have vanished. Since is odd, we have a Fourier sine series. Although only the odd terms in the sine series occur, they fall only as This is similar to the convergence (or lack of convergence ) of harmonic series. Physically this means that our square wave contains a lot of high-frequency components. If the electronic apparatus will not pass these components, our square wave input will emerge more or less rounded off, perhaps as an amorphous blob. Example 7.3.2 Full Wave Rectifier As a second example, let us ask how the output of a full wave rectifier approaches pure direct current (Fig. 7.6). Our rectifier may be thought of as having passed the positive peaks of an incoming sine and inverting the negative peaks. This yields

Since f(t) defined here is even, no terms of the form will appear. (7.37) Since f(t) defined here is even, no terms of the form will appear. Again, from Eqs. (7.11) and (7.12), we have (7.38)

is not an orthogonality interval for both sines and cosines (7.39) Note carefully that is not an orthogonality interval for both sines and cosines together and we do not get zero for even n. The resulting series is (7.40) The original frequency has been eliminated. The lowest frequency oscillation is The high-frequency components fall off as , showing that the full wave rectifier does a fairly good job of approximating direct current. Whether this good approximation is adequate depends on the particular application. If the remaining ac components are objectionable, they may be further suppressed by appropriate filter circuits.

These two examples bring out two features characteristic of Fourier expansion. 1. If f(x) has discontinuities (as in the square wave in Example 7.3.1), we can expect the nth coefficient to be decreasing as . Convergence is relatively slow. If f(x) is continuous (although possibly with discontinuous derivatives as in the Full wave rectifier of example 7.3.2), we can expect the nth coefficient to be decreasing as 3. More generally if f and its first r derivatives are continuous, but the r+1 is not then the nth coefficient will be decreasing as Example 7.3.3 Infinite Series, Riemann Zeta Function As a final example, we consider the purely mathematical problem of expanding (7.41)

by symmetry all For the ’s we have (7.42) (7.43) From this we obtain (7.44)

As it stands, Eq. (7.44) is of no particular importance, but if we set (7.45) and Eq. (7.44) becomes (7.46) or (7.47) thus yielding the Riemann zeta function, , in closed form. From our expansion of and expansions of other powers of x numerous other infinite series can be evaluated.

Some examples of Fourier series… 1. 2. 3. 4. 5. 6.

7.4 Properties of Fourier Series Convergence It might be noted, first that our Fourier series should not be expected to be uniformly convergent if it represents a discontinuous function. A uniformly convergent series of continuous function (sinnx, cosnx) always yields a continuous function. If, however, (a) f(x) is continuous, (b) (c) is sectionally continuous, the Fourier series for f(x) will converge uniformly. These restrictions do not demand that f(x) be periodic, but they will satisfied by continuous, differentiable, periodic function (period of )

Integration Term-by-term integration of the series (7.60) yields (7.61) Clearly, the effect of integration is to place an additional power of n in the denominator of each coefficient. This results in more rapid convergence than before. Consequently, a convergent Fourier series may always be integrated term by term, the resulting series converging uniformly to the integral of the original function. Indeed, term-by-term integration may be valid even if the original series (Eq. (7.60)) is not itself convergent! The function f(x) need only be integrable.

Strictly speaking, Eq. (7.61) may be a Fourier series; that is , if there will be a term . However, (7.62) will still be a Fourier series. Differentiation The situation regarding differentiation is quite different from that of integration. Here thee word is caution. Consider the series for (7.63) We readily find that the Fourier series is (7.64)

Differentiating term by term, we obtain (7.65) which is not convergent ! Warning. Check your derivative For a triangular wave which the convergence is more rapid (and uniform) (7.66) Differentiating term by term (7.67)

which is the Fourier expansion of a square wave (7.68) As the inverse of integration, the operation of differentiation has placed an additional factor n in the numerator of each term. This reduces the rate of convergence and may, as in the first case mentioned, render the differentiated series divergent. In general, term-by-term differentiation is permissible under the same conditions listed for uniform convergence.

COMPLEX FORM OF FOURIER SERIES Recall from earlier that we can write a Fourier series in the complex form in which and From the earlier definitions of we can show that

The multiplication theorem and Parsevals theorem From the complex form of the definition of Fourier series for functions f(x), g(x) of period 2T we can show that Where the c,d’s are the Fourier coefficients of f,g respectively. If we use the repalce the c’d’s by their real form and take f=g we deduce Paresevals theorem , ,